Feb 18, 2016 - Pathways towards instability in financial networks. Marco Bardoscia,1 ..... ska I (2014) Classical mechanics of economic networks. Available at: ...
Pathways towards instability in financial networks Marco Bardoscia,1 Stefano Battiston,2 Fabio Caccioli,3, 4 and Guido Caldarelli5, 6, 1 1 London Institute for Mathematical Sciences, London, UK Department of Banking and Finance, University of Zurich, Switzerland 3 Department of Computer Science, University College London, London, UK 4 Systemic Risk Centre, London School of Economics and Political Sciences, London, UK 5 IMT: Institute for Advanced Studies, Lucca, Italy 6 Institute of Complex Systems CNR, Rome, Italy
arXiv:1602.05883v1 [q-fin.RM] 18 Feb 2016
2
There is growing consensus that processes of market integration and risk diversification may come at the price of more systemic risk. Indeed, financial institutions are interconnected in a network of contracts where distress can either be amplified or dampened. However, a mathematical understanding of instability in relation to the network topology is still lacking. In a model financial network, we show that the origin of instability resides in the presence of specific types of cyclical structures, regardless of many of the details of the distress propagation mechanism. In particular, we show the existence of trajectories in the space of graphs along which a complex network turns from stable to unstable, although at each point along the trajectory its nodes satisfy constraints that would apparently make them individually stable. In the financial context, our findings have important implications for policies aimed at increasing financial stability. We illustrate the propositions on a sample dataset for the top 50 EU listed banks between 2008 and 2013. More in general, our results shed light on previous findings on the instability of model ecosystems and are relevant for a broad class of dynamical processes on complex networks. Keywords: Financial networks, Systemic risk, Stability analisys, Policy making
The systemic risk emerging from networks of financial institutions (in the following banks for brevity) poses significant scientific challenges and comes with prominent policy and societal implications [5]. Banks are interconnected among each other both through direct exposures to bilateral contracts [6, 7] and through indirect exposures to common assets [8, 9]. Here we focus on direct interbank exposures, such as loans extended from one bank to another one, which are usually modeled as directed weighted networks [10–12], while the analysis could be later extended to indirect common exposures [9, 13]. While many factors drive systemic risk, the literature has identified two main channels for the propagation of financial distress through direct exposures. The first is known as illiquidity contagion: If banks anticipate that their counterparties may incur losses, they will try to withdraw their liquid funds from them [14], inducing them, in turn, to withdraw their funds from their own counterparties. Therefore, distress propagates from lenders to borrowers as their liquidity decreases. The second channel is the deterioration of interbank assets: lenders reassess the value of their claims towards borrowers under distress, taking into account the possibility that borrowers will not be able to meet their obligations. Devaluation of assets effectively generates losses for lenders, which can in turn be transmitted to their creditors [6, 15–17]. The propagation of distress is typically mitigated by the intervention of central banks, so that cascades of defaults are rare [18]. Yet, the process of illiquidity contagion is essentially driven by the anticipation of the potential interbank asset deterioration. Therefore, we focus on the latter mechanism only, in line with most of the previous literature [6, 11, 12]. A growing body of work [17, 19–24] carries out stress
tests on the financial system by computing the distribution of losses conditional upon a given pattern of shocks. However, such an approach relies on specific assumptions on the nature of the financial contracts and the distress propagation mechanism. In contrast, here we derive, under mild assumptions, a very general and powerful result on the relation between network structure and stability of the system, and we show how pathways towards instability may emerge in the process of risk diversification. The findings yield specific policy insights that hold valid regardless of the details of financial contracts. We emphasize that, even though here we focus on the interbank network, our results are based on the study of the linear stability around a fixed point and therefore apply to a large class of dynamical processes on networks.
INTERBANK NETWORK
The equity E of a bank, i.e. the difference between its total assets and liabilities, is an important variable in determining the financial health of a bank. In the literature on financial contagion [6, 16, 17], a bank defaults as soon as its equity becomes negative, as it is unlikely that it will be able to repay its debts in full. The ratio between total assets and equity is called leverage and it is a coarse estimate of the riskiness of a bank, as it is related to the maximum loss on the assets that can be absorbed by the equity of the bank. While leverage is usually understood as a single number for each bank, the notion has been recently extended into the concept of leverage matrix [23], whereby leverage is computed with respect to each specific asset class or counterparty. In particular, for a system of n banks here we consider
2 the n × n interbank leverage matrix Λ, whose elements Λij are equal to the ratio between the nominal exposure of bank i towards bank j and the equity of bank i. The total P interbank leverage of bank i is simply equal to `i = j Λij . In fact, we will consider an adjusted ˆ ij = Λij (1 − ρj ), where ρj interbank leverage matrix Λ is the recovery rate of bank j, i.e. the fraction of its interbank assets recovered by creditors in case of default. Finally, let us denote the relative equity losses of bank i as hi (t) = (Ei (0) − Ei (t))/Ei (0). Starting from basic principles of financial accounting and under mild assumptions on the type of financial contracts among banks, we show that the relative equity loss of bank i must be a function of the relative equity loss of its counterparties, as well as function of the leverage matrix Λij , according to the following dynamics: P ˆ hi (t + 1) = hi (1) + j Λ ij p(hj (t)), where p is the default probability of counterparty j as a function of its relative equity loss (see SI Appendix for the details). Even though the function p is, in general, complicated and nonlinear, the response of the system to a small initial shock can be characterized by linear stability analysis around a fixed point. This implies that the system is unstable if and only if λmax , the largest eigenvalue of the Jaco˜ ij = Λ ˆ ij p0 (hj (0)), is larger than bian of the dynamics Λ j one. In the following, without loss of generality, we assume p0j (hj (0)) = 1, ρj = 0 for all j, and we denote ˜ = Λ, although all the results hold in the more general Λ case. Notice that the presence of a recovery rate pushes the system towards stability, as the largest eigenvalue of Λ is larger than or equal to the largest eigenvalue of ˆ Notice also that under the further assumption that Λ. p(h) = h, we would recover the extension of the DebtRank dynamics [17, 23, 24] that includes a recovery rate (see SI Appendix). Despite the considerable body of work on financial contagion, since there is no simple relationship between the topology of a network and λmax , the study of stability has been seldom carried out [25] in this context. Here, we show two important effects pertaining financial instability that had remained uncovered so far and could have profound policy implications. First, even if the individual leverage of banks does not increase, a financial system can turn from stable to unstable as the number of banks increases (i.e. the number of nodes in the network grows larger) like during a process of market integration. Second, even if the individual leverage of banks does not increase, a financial system can become unstable as the number of contracts among banks increases (i.e. the number of edges in the network increases) like during a process of risk diversification. Notably, in both cases instability appears despite the fact that the assessment that each bank makes of its own risk profile does not change, because individual leverage levels remain constant. This means that market integration and risk diversification can make the system as a whole unstable. These results do not imply that such processes are detrimental
per se, but that financial policies focusing only on individual banks, also known as micro-prudential policies, can have the opposite effect of increasing financial instability if they do not consider the system as a whole. As it will be clear further below, the origin of instability lies in the fact that in both processes banks get increasingly involved in multiple cycles (i.e. closed chains) of contracts. Our results suggest to include the eigenvalue analysis of the leverage matrix among the tools of financial stability.
EMERGENCE OF INSTABILITY
The relation between λmax and interbank leverage across banks becomes simple if all banks have the same interbank leverage or if the interbank network is a large Erd˝os-R´enyi graph [27]. In the first case, via the PerronFrobenius theorem, λmax is bounded by the smallest and largest sum over the columns of the interbank leverage matrix, i.e. precisely by the smallest and largest interbank leverages. Hence, if all banks have the same interbank leverage `, it must be also equal to λmax . The second case is similar to the May-Wigner theorem about the instability of model ecosystems [28] in which species interact through a large Erd˝os-R´enyi graph. The main difference is that in our case interactions between banks are described by the leverage matrix Λ, which is non-negative, while the interactions between species in ecosystems are described by a matrix whose elements can have unspecified sign. In the SI AppendixPwe prove that, for n → ∞, in this case λmax → ` = i `i /n = P i,j Λij /n, the average interbank leverage across banks. Therefore, in both cases the system is unstable whenever ` > 1. When relaxing either of the two assumptions (homogeneity of leverage, or large size together with randomness of the graph), finer details of the network structure become important. For instance, because the theorem only holds in the limit of large size graphs, there exist small Erd˝os-Renyi graphs that are stable although they have ` > 1. An example of a small size network that is extremely important for policy is the the network of the Global Systemically Important Banks [29], comprising about 30 banks. Suppose to start from a small and stable Erd˝os-Renyi graph with ` > 1 and to connect more banks to the network (by keeping ` and the number of contracts per bank constant). Eventually, the system will grow large enough to become unstable because the theorem will have to hold in the limit of large graphs (see SI Appendix for an example). This is an example of a previously unreported phenomenon that we call pathways to instability, i.e. the existence of trajectories in the space of graphs along which financial networks turn from stable to unstable, although at each point along the trajectory banks individually satisfy constraints that would apparently make them individually stable. In general, the system is unstable if and only if there exists an unstable strongly connected component (i.e. a
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FIG. 1. For illustrative purposes we perform the stability analysis of the two paradigmatic interbank network architectures depicted above. The first is a “butterfly” graph, while the second example has a core-periphery topology: nodes 1, 2, 3, and 4 form a complete core, with the remaining nodes having either only incoming or outgoing edges to the core. For simplicity we set all non-zero elements of the interbank leverage matrix equal to w, implying that the largest single exposure policy is implemented whenever w < 1. We plot the average interbank leverage (green line) and λmax , the largest eigenvalue of the interbank leverage matrix (red line), as functions of the parameter w. The green region corresponds to an average interbank leverage smaller than one, the blue region to the largest single exposure smaller than the corresponding equity, while the unstable region is highlighted in red. In both cases there exists a region (shadowed in the figure) in which the following three properties hold: i) the average interbank leverage is larger than one, ii) the largest single exposure is smaller than the corresponding equity, and yet iii) the network is unstable. Slight modifications of the above examples can also account for tighter constraints on the largest single exposure. For example, even requiring that the largest single exposure is smaller than 15% of the equity (as requested in [26]) is not enough to avoid instability in a core-periphery topology with eight nodes in the core.
directed subgraph in which each node is reachable indirectly by any other). The Perron-Frobenius theorem only guarantees that the largest eigenvalue of a strongly connected component is between the minimum and the maximum interbank leverage across banks. Hence, a sufficient condition for instability (stability) is that the interbank leverage of all banks is larger (smaller) than one. However, for the years from 2008 to 2013, the smallest interbank leverage of European banks is very close to zero, while the 95th percentile of its distribution is between 2.5 and 6, meaning that the Perron-Frobenius bounds are not informative enough on the largest eigenvalue, and we need to look more closely at the topology of the network. For instance, for graphs without cycles (i.e. directed acyclic graphs, DAGs) λmax is always equal to
zero, implying that the presence of cycles is a necessary condition for instability (although not sufficient). Intuitively, a cycle amplifies distress propagation if the product of the weights of its edges is larger than one (we refer to this as an individually unstable cycle). Interestingly, a policy recommendation included in Basel III Accords [26] encourages banks to have the largest single exposure smaller than a fraction of their equity, so that Λij < 1 for all i, j. The policy is thus effective in avoiding this source of instability. However, the presence of individually unstable cycles, although sufficient, is not necessary for instability. Consider the two examples in Fig. 1. In particular, the second is a simple case of core-periphery network architecture, a frequently observed pattern in empirical interbank data
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FIG. 2. Stability of the network of the top 50 listed European banks using data from their 2013 balance sheets. Trajectories have been built according to the following protocol: We start from a random DAG, i.e. a network with no cycles, which is therefore stable. We then increase the density of edges in the network by randomly adding one edge at a time, until the network is complete. Every time a new edge is added, we re-balance the interbank exposures so that the interbank leverages are always consistent with the real balance sheets. We compute λmax , the largest eigenvalue of the interbank leverage matrix, to assess the stability of the network: if λmax > 1 (< 1) the system is unstable (stable). We repeat the whole procedure 100 times. We show the contour of all trajectories and highlight a few of them. The first crossing region (in semi-transparent blue), spans the interval of densities of edges across which the networks become unstable for the first time, meaning that combined unstable cycles appear. We can see that densities as low as 3% are sufficient to reach instability. We also plot the average interbank leverage (dashed blue line) for reference.
[30]. In both cases, not only the largest single exposure policy is implemented, but (depending on the value of the parameter w) the average interbank leverage can be smaller than one. These two conditions could intuitively suggest that the system is stable. Yet, λmax is larger than one and the system is unstable. The reason is that there are banks involved in multiple cycles. More precisely, a sufficient condition for having λmax > 1 is that there exist two integers i, k such that (Λk )ii > 1, i.e. that there exists a bank i such that the sum, over all the cycles of length k from i to itself, of the products of the elements of the interbank leverage matrix along each of such cycles is larger than one (we refer to this as a combined unstable cycle). For instance, in the first example of Fig. 1, (Λ3 )11 is larger than one for w > 2−1/3 , and thus there is a range of values where the system is unstable even if the largest single exposure policy is implemented and the average interbank leverage is smaller than 1. The sufficient condition for instability stated above has important consequences for regulations intended to promote financial stability. Take the case of a bank having a given interbank leverage and at least one exposure larger than its equity. If now the bank is required to implement the largest single exposure policy and it wants to keep its interbank leverage unchanged, it might have to increase the number of its counterparties. On the one hand, this is beneficial because it reduces the exposures towards individual counterparties. On the other hand, it might be detrimental as it could contribute to the creation of new
cycles that, even though might be individually stable, are part of a combined unstable cycle. Therefore, a recommendation that targets stability in terms of individual banks it can actually lead to instability because it neglects the systemic effect of cycles. More in general, increasing the number of contracts in the system is the source of a second type of pathway towards instability. As a empirical illustration of this phenomenon, we consider the balance sheets of the top 50 listed banks in the European Union obtained from the Bankscope dataset. We simulate a process in which banks gradually increase the degree of risk diversification by continuously creating exposures towards additional counterparties. In particular, by adding edges, we build trajectories in the space of interbank networks whose initial configuration is a random DAG (hence stable) and whose final configuration is a complete graph. Every time a new edge is added, interbank exposures are redistributed so that the network is always consistent with the original balance sheets and interbank leverages of all banks do not change. For simplicity, we set the recovery rate ρ equal to zero. We find that the banking system is unstable once its graph is complete, but actually instability kicks in much earlier, when the fraction of existing contracts over all the possible ones is as low as 3% (see Fig. 2 for 2013 balance sheets, and SI Appendix for other years). Moreover, from Fig. 2 we see that trajectories of λmax can be not monotonic and that the critical line can be crossed multiple times, meaning
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FIG. 3. Toy model of an interbank network that oscillates between stability and instability. Going from a to e we add one or more edges every time, always redistributing the weights so that interbank leverages do not change. Added edges are green, while modified edges are red. The initial network a is a DAG, hence λmax = 0, and for simplicity all edges have the same (e) (d) weight w. Suppose that w is chosen such that λmax < 1 < λmax . We then have that network b is stable, even tough a cycle has appeared. The further addition of one more cycle makes network c unstable. Network d becomes stable again after the addition of two edges, and finally network e is again unstable.
that the system sways between stability and instability, before finally settling into an unstable state. In Fig. 3 we provide a stylized example that helps to connect such changes in the stability of the system to changes in the topology of the network. We start from a DAG, initially setting all non-zero elements of the interbank leverage matrix equal to w. We then add one edge at a time, always distributing the interbank leverage of each bank uniformly among the neighboring (borrowing) banks. λmax increases every time a new cycle appears in the system. In contrast, λmax decreases whenever a new edge does not lead to the appearance of a new cycle. Intuitively, this behavior can be explained in the following way. On the one hand, whenever a new cycle appears the possibility for the system to amplify shocks increases. On the other hand, whenever the addition of a
new edge does not lead to the creation of a new cycle, the weights of those edges that are part of existing cycles become smaller because interbank leverages are constantly re-balanced, decreasing the ability of those cycles to amplify shocks.
CONCLUSIONS
By providing a simple and rigorous mathematical explanation of how network effects arise our results shed new light on the tension between the two main approaches to financial stability: the so-called microprudential one, focused on ensuring the stability of individual banks, and the macroprudential ones, targeted to the stability of the whole financial system.
6 We provide examples of sufficient conditions for the onset of instability: when banks establish contracts among each other without taking into account what their counterparties do, they will eventually become even unintentionally part of multiple cycles of contracts, which altogether amplify the effects of shocks. The recovery rate plays an important role, as it impacts directly the critical value of the largest eigenvalue. In turn, the recovery rate can be at least in part controlled with certain financial and monetary policies since it depends on both the quality of the collateral (in case of secured lending) and on the liquidity of the asset markets. Overall, our findings suggest that financial stability policies need to carefully consider network effects. This can be achieved by computing the largest eigenvalue of the interbank leverage matrix and by comparing it with estimates of the recovery rate. More specifically, we show the existence of two processes that define trajectories in the space of network configurations, which drive financial networks from a stable to an unstable regime. The former consists of implementing processes of market integration (i.e. increasing the number of financial institutions) in a growing interbank network with interbank leverage larger than one. The latter consists of increasing the number of contracts among financial institutions. In both cases the risk profile of individual banks (measured by the interbank leverage) does not change, and therefore the emergence of instability is purely related to the structure of the network. This suggests that policies targeted at ensuring financial stability by lowering the risk of individual banks without taking into account the network effects can in fact lead to a higher systemic risk. The relation between the topology of a network and the behaviour of dynamical processes taking place on the network is a fundamental problem common to a broad
class of complex systems. Here we focus on financial contagion in a interbank network, but our results apply to any system in which the Jacobian of the dynamics is proportional to the weighted adjacency matrix (with non-negative entries) encoding the mutual interactions between the constituents of the system. In fact, our results expressed in terms of interbank leverages (sums over columns of the interbank leverage matrix) can be rephrased in terms of what in a more general setting are known as strengths of nodes (sums over columns of a weighted adjacency matrix). As a consequence, the existence of pathways to instability could in principle be observed in diverse contexts ranging from ecosystems to social networks.
[1] Battiston S et al. (2016) Complexity theory to support public policy and financial regulation. Science, in press. [2] Eisenberg L, Noe TH (2001) Systemic Risk in Financial Systems. Management Science 47(2):236–249. [3] Poledna S, Molina-Borboa J, van der Leij M, MartinezJaramillo S, Thurner S (2015) Multi-layer network nature of systemic risk in financial networks and its implications. Journal of Financial Stability 20:70–81. [4] Elsinger H, Lehar A, Summer M (2006) Risk Assessment for Banking Systems. Management Science 52(9):1301– 1314. [5] Caccioli F, Shrestha M, Moore C, Doyne Farmer J (2014) Stability analysis of financial contagion due to overlapping portfolios. Journal of Banking & Finance 46:233– 245. [6] Allen F, Gale D (2000) Financial contagion. Journal of political economy 108(1):1–33. [7] Acemoglu D, Ozdaglar A, Tahbaz-Salehi A (2015) Systemic Risk and Stability in Financial Networks. American Economic Review 105(2):564–608.
[8] Elliott M, Golub B, Jackson MO (2014) Financial Networks and Contagion. American Economic Review 104(10):3115–3153. [9] Corsi F, Marmi S, Lillo F (2015) When micro prudence increases macro risk: The destabilizing effects of financial innovation, leverage, and diversification. Operations Research, in press. Available at: http://ssrn.com/abstract=2278298. [10] Gai P, Haldane A, Kapadia S (2011) Complexity, concentration and contagion. Journal of Monetary Economics 58(5):453–470. [11] Haldane AG, May RM (2011) Systemic risk in banking ecosystems. Nature 469(7330):351–5. [12] Furfine CH (2003) Interbank Exposures: Quantifying the Risk of Contagion. Journal of Money, Credit & Banking 35(1):111–129. [13] Battiston S, Puliga M, Kaushik R, Tasca P, Caldarelli G (2012) DebtRank: Too Central to Fail? Financial Networks, the FED and Systemic Risk. Scientific Reports 2:541.
ACKNOWLEDGMENTS
We are grateful to Joseph E. Stiglitz for his comments on the manuscript. We also acknowledge Marco D’Errico for the discussions on the notion of interbank leverage and for sharing the banks’ balance sheet data used in the empirical application. MB, SB, and GC acknowledge support from: FET Project SIMPOL nr. 610704, FET Project DOLFINS nr. 640772, and FET IP Project MULTIPLEX nr. 317532. FC acknowledges support of the Economic and Social Research Council (ESRC) in funding the Systemic Risk Centre (ES/K002309/1). SB acknowledges the Swiss National Fund Professorship grant nr. PP00P1-144689. GC acknowledges also EU projects SoBigData nr. 654024 and CoeGSS nr. 676547. The banks’ balance sheet data come from the Bureau van Dijk Bankscope’s database accessible to the Univ. of Zurich. The authors declare no competing interests.
7 [14] Glasserman P, Young P, Young HP (2015) How Likely is gation. PLoS One 10(7):e0134888. Contagion in Financial Networks. Journal of Banking & [21] Markose S, Giansante S, Shaghaghi AR (2012) Too InterFinance 50:383–399. connected To FailFinancial Network of US CDS Market: [15] Amini H, Cont R, Minca A (2012) Stress testing the reTopological Fragility and Systemic Risk. Journal of Ecosilience of financial networks. International Journal of nomic Behavior & Organization 83(3):627–646. Theoretical and Applied Finance 15(1):1250006. [22] Basel Committee on Banking Supervision (2014) Super[16] Gai P, Kapadia S (2010) Contagion in financial networks. visory framework for measuring and controlling large exProceedings of the Royal Society A: Mathematical, Physposures. http://www.bis.org/publ/bcbs283.pdf; accessed ical and Engineering Sciences 466(2120):2401–2423. 20-January-2016. [17] Dehmamy N, Buldyrev SV, Havlin S, Stanley HE, Voden[23] Bollob´ as B (2001) Random Graphs. (Cambridge Universka I (2014) Classical mechanics of economic networks. sity Press, Cambridge), 2 edition. Available at: http://arxiv.org/abs/1410.0104. [24] May RM (1972) Will a large complex system be stable? [18] Montagna M, Lux T (2014) Contagion risk in the Nature 238:413–414. interbank market: A probabilistic approach to cope [25] Financial Stability Board (2014) 2014 update with incomplete structural information. Available at: of list of global systemically important banks. https://www.econstor.eu/dspace/bitstream/10419/102271/1/wp-http://www.fsb.org/wp-content/uploads/r 141106b.pdf; accessed 20-January-2016. 08.pdf. [26] Craig B, Von Peter G (2014) Interbank tiering and [19] Battiston S, Caldarelli G, D’Errico M, Gurciullo S (2015) money center banks. Journal of Financial IntermediaLeveraging the network: a stress-test framework based tion 23(3):322–347. on DebtRank. Statistics & Risk Modeling, forthcoming. Available at: http://ssrn.com/abstract=2571218. [20] Bardoscia M, Battiston S, Caccioli F, Caldarelli G (2015) DebtRank: A Microscopic Foundation for Shock Propa-
S1
Supporting Information Appendix LINEAR APPROXIMATION OF CONTAGION DYNAMICS
In this section we derive, under general mild assumptions, the linear approximation of distress propagation in interbank networks and we show how the stability of the system is related to the interbank leverage matrix. The first important ingredient is the balance-sheet consistency at all times t. The balance sheet of a bank is composed by assets and liabilities. The former have positive economic value (e.g. loans towards customers or towards other banks, stocks, derivatives, real estate), while the latter have negative economic value (e.g. deposits, debits towards other banks). In both cases, we distinguish between interbank and external assets or liabilities. Interbank assets (liabilities) are credits (debits) of banks towards other banks, while we call external all other assets and liabilities. We denote by Aij (t) the value at time t of a loan from bank i to bank j, and by Lji (t) the corresponding liability. External assets E and liabilities of bank i at time t are denoted by AE i (t) and Li (t), respectively. Finally, the equity Ei (t) of bank i at time t is defined as the difference between its assets and liabilities: E Ei (t) = AE i (t) − Li (t) +
n X j=1
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(S1)
We follow the assumption, common in the literature, that a bank defaults if its equity becomes negative. The rationale is that a bank with more debts that assets is not able to meet its obligations towards its creditors even by liquidating its entire pool of assets. The second important ingredient is that the value of interbank assets change over time, as lenders constantly reassess their value based on the probability of default of their counterparties. Nevertheless, the actual amount of outstanding debt of borrowers does not change, i.e. liabilities always keep their face value. If bank j does not default its lender i will get Aij (0), the face value of the loan, while if bank j defaults i recovers the (smaller) amount Rij . We assume that the value of interbank loans is assessed “fairly”, i.e. it is equal to the expected repayment: Aij (t) = Aij (0)(1 − pj (t − 1)) + Rij pj (t − 1) ,
(S2)
where pj (t) is the probability of default of bank j at time t. The time delay from the r.h.s. and the l.h.s. of (S2) accounts for the time needed for the information about the probability of default of borrowers to be incorporated into the assessment of lenders. Let us explicitly note that (S2) is consistent with assuming that pj (0) = 0. The scenario we have in mind is to initially stress the system via an exogenous shock to external assets, i.e. E E AE i (0) → Ai (1) < Ai (0). Balance sheet consistency (S1) implies that such shock will result in losses in equity. We assume that no additional cash flow (neither positive nor negative) enters the system subsequently. By assuming that the probability of default of a bank is an arbitrary function of its equity, equity losses will translate in changes in probabilities of default, and, via (S2), into a re-evaluation of interbank assets. This, in turn, will lead (again via (S1)) to a change in equity. In subsequent rounds external assets do not change and propagation of shocks continues only by iterating such dynamic through the interbank channel. As a consequence, two terms contribute to the loss in equity of bank i between time 0 to time t: the loss in external assets between time 0 and time 1 and the loss in interbank assets up to time t: Ei (0) − Ei (t) =
AE i (0)
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ˆ ij pj (hj (t − 1)) , Λ
(S6)
S2 where we also highlight the explicit dependence of the probability of default of bank j on its equity, and therefore on hj . We stress that the assumptions made so far (balance sheets consistency, fair re-evaluation of interbank assets, probability of default as a generic function of the equity) can be considered accounting first principles. By further assuming that pj (hj (t)) is differentiable in hj (0): pj (hj (t)) ' pj (hj (0)) + p0j (hj (0)) [hj (t) − hj (0)] = p0j (hj (0))hj (t) ,
(S7)
where we have used that hj (0) = 0. By plugging (S7) into (S6) and by defining: ˜ ij = Aij (0) − Rij p0j (hj (0)) , Λ Ei (0)
(S8)
we can finally write hi (t) = hi (1) +
n X j=1
˜ ij hj (t − 1) . Λ
(S9)
From the point of view of the stability of the system it is useful to compute hi (t + 1) − hi (t), which leads to: n X ˜ ij [hj (t) − hj (t − 1)] , Λ (S10) hi (t + 1) = min 1, hi (t) + j=1
where the min simply ensures that relative equity losses cannot become larger than one, i.e. that equities do not become negative. However, equities stay positive in-between defaults and, by introducing ∆h(t) = h(t) − h(t − 1), (S10) can be written in matrix form: ˜ ∆h(t + 1) = Λ∆h(t) ,
(S11)
which is a linear map with fixed point ∆h = 0. The fixed point is stable if the modulus of the largest eigenvalue λmax ˜ is smaller than one. If λmax > 1, shocks will be amplified and at least one bank will default. of Λ A further simplifying assumption is that Rij , the amount recovered by the lender bank i in case of default of the borrower bank j, is a fraction ρj of the face value Aij (0): Rij = ρj Aij (0) .
(S12)
From (S10) and (S12) we can see that ρj effectively dampens the losses suffered by lenders of bank j at all time steps, hence we call it recovery rate. Eqs. (S5) and (S8) become: ˆ ij = Λij (1 − ρj ) Λ
(S13a)
˜ ij = Λij (1 − ρj )p0 (hj (0)) , Λ j
(S13b)
where Λij =
Aij (0) , Ei (0)
(S14)
is the interbank leverage matrix. From (S13b) we can make two observations. First, the structure of the network ˜ i.e. the couple of banks between which there is an edge, is equal to the structure of the network underlying underlying Λ, Λ. Second, a difference from bank to bank in the functions pj is completely indistinguishable from heterogeneous recovery rates, by defining effective recovery rates one can absorb the factors p0j (hj (0)). Choosing the equity loss as probability of default:
implies that
p0j (hj (0))
pj (hj ((t)) = hj (t) ,
(S15)
˜ =Λ ˆ = Λ(1 − ρ) , Λ
(S16)
= 1, so that:
where ρ is a matrix whose j-th element on the diagonal is equal to ρj and with off-diagonal elements equal to zero. In this case (S7) is not an approximation, meaning that the dynamics (S10) and (S11) are exact and closely resemble the ˜ → Λ we obtain a generalized DebtRank [S1] linear dynamic ∆h(t + 1) = Λ∆h(t). By performing the replacement Λ DebtRank that includes recovery.
S3 ADDING NODES
The crucial thereom that we will exploit is due to Silverstein [S4] (Theorem 1.2). In a nutshell, let Λ be a n × n matrix whose entries are random i.i.d. variables with mean µ > 0 and finite forth moment. For sufficiently large n, the largest eigenvalue λmax of Λ is: λmax =
1X Λij + O(n−1/2 ) . n i,j
(S17)
We will now specify the results of the theorem in the case in which the matrix Λ is the weighted adjacency matrix of a random graph. We consider Erd˝ os-Renyi graphs in which Λij = Cij Wij , with Cij ∈ {0, 1} and Wij ∈ R+ . The variables Cij determine if an edge is present or not and have the bimodal distribution ρ(Cij ) = pδ(Cij − 1) + (1 − p)δ(Cij ). The variables Wij are the weights associated to the edges and we leave their distribution unspecified (as long as the forth moment is finite). P We start with the case in which the network is not sparse, i.e. the case in which the average degree k¯ ≡ ij Cij /n is k¯ ' O(n), or equivalently p ' O(1) (in the sense that P it does not scale with n). Let us define the variables Xi , i = 1, . . . n, as the sums only over columns of Λ, i.e. Xi = j Cij Wij . As Cij and Wij are independent, we have:
The next step is to compute normally distributed with
P
i
hXi i = nhCij ihWij i = nphWij i
(S18a)
var Xi = n var(Cij Wij ) = n phWij2 i − p2 hWij i2 .
(S18b)
Xi /n. As Xi are i.i.d. with finite variance, using (S17) we have that λmax will be
hλmax i = var λmax =
1 nhXi i = nphWij i n
1 n var Xi = phWij2 i − p2 hWij i2 , 2 n
(S19a)
(S19b)
√ meaning that the relative fluctuation is var λmax /hλmax i ' 1/n. In the case in which the graph is sparse, i.e. k¯ ' O(1) and p ' 1/n we know that the degree of each node has a ¯ As a consequence, Xi will have a compound Poisson distribution with Poisson distribution with mean k.
If we now compute the first two moments of
P
i
¯ ij i hXi i = khW
(S20a)
¯ 2i. var Xi = khW ij
(S20b)
Xi /n we find that:
hλmax i =
1 ¯ ij i nhXi i = khW n
(S21a)
¯ 2i khW 1 ij n var X = , (S21b) i n2 n √ √ meaning that the relative fluctuation is var λmax /hλmax i ' 1/ n. Moreover, we can see that the fluctuation on hλmax i is of the same order of the correction in (S17), therefore we are not able to compute the distribution of λmax in this case. In the previous derivation we assumed that all entries of the interbank leverage matrix are i.i.d., which is not entirely true. In fact, in our networks a bank cannot extend a loan to itself, meaning that there are no loops (cycles of length one), i.e. the diagonal of the weighted adjacency matrix is filled with zeros. To compute the relative correction on hλmax i it will suffice to note that if λ is an eigenvalue of a matrix M , λ − a is an eigenvalue of the matrix M − aI. As a consequence, in the case of sparse graphs, we have that hλmax i = nphWij i − phWij i = (n − 1)phWij i. Since for ¯ ij i. In the case of sparse graphs the correction is graphs without loops k¯ = (n − 1)p, we have that hλmax i = khW already accounted for in (S21a), provided that the correct value of k¯ is used. var λmax =
1.6
largest eigenvalue average interbank leverage density
1.4
1.6 1.4
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0 0
20
40
60 t
80
100
density
largest eigenvalue / average interbank leverage
S4
0.0 120
FIG. S1. Example of growth process in which a stable network with average interbank leverage larger than one becomes unstable as new banks are added to the system. We stress that the crossing to the unstable regime is genuinely driven by the fact that fluctuations in the asymptotic distribution of λmax shrink as n becomes larger: in fact the density of edges in the network stays roughly constant. Here the initial network has n = 20 and the weight distribution is exponential.
¯ ij i, as n → ∞, but with different relative fluctuations. It is worth noting In both cases we have that λmax = khW ¯ ij i is precisely the average interbank leverage `. Therefore, for that, when Λ is the matrix of interbank leverage, khW n → ∞, if ` > 1 the system will be unstable, while if ` < 1 it will be stable. However, if n is not large, fluctuations are relevant, and a system can be stable even if ` > 1, and vice versa. We now provide an example of how adding nodes to such a network can make the system unstable. We start by randomly generating an Erd˝os-Renyi graph with given p and using an exponential distribution of weights with mean hWij i, so that ` > 1, stopping as soon as we find a stable graph. We then proceed to add a new node at a time, by preserving the property that all entries of ¯ constant. In fact, if we devised the weighted adjacency matrix are i.i.d. and by keeping the density of edges (i.e. k) ¯ a growth process in which k increases, the system would trivially become unstable. We use the following algorithm. Let n be the number of nodes before the addition of a new node i. (i) We randomly form edges from node i and each of the other n nodes with probability p; (ii) we draw a weight from the weight distribution for each of the new outcoming edges from i: (iii) we rescale such weights multiplying them by (n − 1)/n; (iv) we randomly form edges from each of the other n nodes to node i with probability p; (v) we draw a weight from the weight distribution for each of the new incoming edges for i; (vi) we rescale the weights of all edges starting from the new neighbors of i (including the ones towards node i) so that the sum of all weights of the edges coming out from those nodes do not change after the addition of node i. In Fig. S1 we see a realization of such process in which both the density of edges and the average interbank leverage are roughly constant, while λmax becomes larger than one, driving the system towards the instability. Let us note that such algorithm is designed to keep all interbank leverages constant. However, the probability distribution of single entries of the interbank leverage matrix may vary from a step of the algorithm to the next one. We have checked that the simpler variant in which one keeps the probability distribution of single entries constant and the interbank leverage constant only on average yields the same results.
ADDING EDGES
For completeness, we show in Fig. S2 here the analogous of Fig. 3 of the main text for the years between 2008 and 2012. We also include the year 2013 (already in Fig. 3 of the main text) for reference.
S5 8
5
7 4
6
3 λmax
λmax
5 4
2
3 2
1
1 0
10−2
10−1 density
0
100
6
10−2
10−1 density
100
10−2
10−1 density
100
8 7
5
6 5 λmax
λmax
4 3
4 3
2
2 1 0
1 10−2
10−1 density
0
100
4.0
3.5
3.5
3.0
3.0
2.5 λmax
λmax
2.5 2.0 1.5
1.5 1.0
1.0
0.5
0.5 0.0
2.0
10−2
10−1 density
100
0.0
10−2
10−1 density
100
FIG. S2. Analogous of Fig. 3 of the main text for years from 2008 to 2013. 2008: upper left, 2009: upper right, 2010: mid left, 2011: mid right, 2012: bottom left, 2013: bottom right. For λmax < 1 the interbank network is stable (green region), while for λmax > 1 it is unstable (red region). For comparison we also plot (dashed blue line) the average interbank leverage.
S6
[S1] Bardoscia M, Battiston S, Caccioli F, Caldarelli G (2015) DebtRank: Microscopic foundation of shock propagation. PLoS ONE 10(6): e0130406. [S2] Furfine C H (2003) Interbank Exposures: Quantifying the Risk of Contagion. Journal of Money, Credit and Banking 35: 111 – 128. [S3] Eisenberg L, Noe T H (2001) Systemic Risk in Financial Systems. Management Science 47(2): 236 – 249. [S4] Silverstein J W (1994) The spectral radii and norms of large dimensional non-central random matrices. Stochastic Models 10(3): 525 – 532. [S5] Battiston S et al. (2016) Complexity theory to support public policy and financial regulation. Science, in press. [S6] Eisenberg L, Noe TH (2001) Systemic Risk in Financial Systems. Management Science 47(2):236–249. [S7] Poledna S, Molina-Borboa J, van der Leij M, Martinez-Jaramillo S, Thurner S (2015) Multi-layer network nature of systemic risk in financial networks and its implications. Journal of Financial Stability 20:70–81. [S8] Elsinger H, Lehar A, Summer M (2006) Risk Assessment for Banking Systems. Management Science 52(9):1301–1314. [S9] Caccioli F, Shrestha M, Moore C, Doyne Farmer J (2014) Stability analysis of financial contagion due to overlapping portfolios. Journal of Banking & Finance 46:233–245. [S10] Allen F, Gale D (2000) Financial contagion. Journal of political economy 108(1):1–33. [S11] Acemoglu D, Ozdaglar A, Tahbaz-Salehi A (2015) Systemic Risk and Stability in Financial Networks. American Economic Review 105(2):564–608. [S12] Elliott M, Golub B, Jackson MO (2014) Financial Networks and Contagion. American Economic Review 104(10):3115– 3153. [S13] Corsi F, Marmi S, Lillo F (2015) When micro prudence increases macro risk: The destabilizing effects of financial innovation, leverage, and diversification. Operations Research, in press. Available at: http://ssrn.com/abstract=2278298. [S14] Gai P, Haldane A, Kapadia S (2011) Complexity, concentration and contagion. Journal of Monetary Economics 58(5):453– 470. [S15] Haldane AG, May RM (2011) Systemic risk in banking ecosystems. Nature 469(7330):351–5. [S16] Furfine CH (2003) Interbank Exposures: Quantifying the Risk of Contagion. Journal of Money, Credit & Banking 35(1):111–129. [S17] Battiston S, Puliga M, Kaushik R, Tasca P, Caldarelli G (2012) DebtRank: Too Central to Fail? Financial Networks, the FED and Systemic Risk. Scientific Reports 2:541. [S18] Glasserman P, Young P, Young HP (2015) How Likely is Contagion in Financial Networks. Journal of Banking & Finance 50:383–399. [S19] Amini H, Cont R, Minca A (2012) Stress testing the resilience of financial networks. International Journal of Theoretical and Applied Finance 15(1):1250006. [S20] Gai P, Kapadia S (2010) Contagion in financial networks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466(2120):2401–2423. [S21] Dehmamy N, Buldyrev SV, Havlin S, Stanley HE, Vodenska I (2014) Classical mechanics of economic networks. Available at: http://arxiv.org/abs/1410.0104. [S22] Montagna M, Lux T (2014) Contagion risk in the interbank market: A probabilistic approach to cope with incomplete structural information. Available at: https://www.econstor.eu/dspace/bitstream/10419/102271/1/wp-08.pdf. [S23] Battiston S, Caldarelli G, D’Errico M, Gurciullo S (2015) Leveraging the network: a stress-test framework based on DebtRank. Statistics & Risk Modeling, forthcoming. Available at: http://ssrn.com/abstract=2571218. [S24] Bardoscia M, Battiston S, Caccioli F, Caldarelli G (2015) DebtRank: A Microscopic Foundation for Shock Propagation. PLoS One 10(7):e0134888. [S25] Markose S, Giansante S, Shaghaghi AR (2012) Too Interconnected To FailFinancial Network of US CDS Market: Topological Fragility and Systemic Risk. Journal of Economic Behavior & Organization 83(3):627–646. [S26] Basel Committee on Banking Supervision (2014) Supervisory framework for measuring and controlling large exposures. http://www.bis.org/publ/bcbs283.pdf; accessed 20-January-2016. [S27] Bollob´ as B (2001) Random Graphs. (Cambridge University Press, Cambridge), 2 edition. [S28] May RM (1972) Will a large complex system be stable? Nature 238:413–414. [S29] Financial Stability Board (2014) 2014 update of list of global systemically important banks. http://www.fsb.org/wpcontent/uploads/r 141106b.pdf; accessed 20-January-2016. [S30] Craig B, Von Peter G (2014) Interbank tiering and money center banks. Journal of Financial Intermediation 23(3):322– 347.
